In graph theory, a **tree** is a type of graph that has specific properties. Here are the key characteristics that define a tree: 1. **Acyclic**: A tree is acyclic, meaning it does not contain any cycles. In other words, there is no way to start at one vertex, travel along the edges, and return to the same vertex without retracing steps.

A **cactus graph** is a special type of graph in graph theory with a specific structural property. A cactus graph is defined as a connected graph in which any two cycles have at most one vertex in common. In simpler terms, while a cactus can have multiple cycles, these cycles cannot intersect in more than one vertex, meaning that their intersections (if any) do not create complex overlapping structures.

A comparability graph is a type of graph that arises in the field of graph theory, specifically in the study of ordered sets (partially ordered sets or posets). In a comparability graph, the vertices represent elements of a partially ordered set, and there is an edge between two vertices if and only if the corresponding elements are comparable in the poset. This means one element is either less than or greater than the other according to the ordering.

A **dense graph** is a type of graph in which the number of edges is close to the maximal number of edges that can exist between the vertices. More formally, a graph is considered dense if the ratio of the number of edges \( E \) to the number of vertices \( V \) squared, \( \frac{E}{V^2} \), is relatively large.

A **dually chordal graph** is a type of graph that has specific structural properties related to both its vertices and cycles. The term "dually chordal" arises in the context of vertex or edge properties. 1. **Chordal Graph**: - A graph is called **chordal** if every cycle of length four or more has a chord. A chord is an edge that is not part of the cycle but connects two vertices of the cycle.

In the context of mathematical logic and set theory, particularly in the area of model theory and set-theoretic topology, a **forcing graph** is not a standard term. However, it may refer to concepts related to forcing conditions in the context of set theory. **Forcing** is a technique introduced by Paul Cohen in the 1960s.

A scale-free network is a type of network characterized by a particular property in its degree distribution. In such networks, the distribution of connections (or edges) among the nodes follows a power law, which means that a few nodes (often referred to as "hubs") have a very high number of connections, while the majority of nodes have relatively few connections.

In graph theory, a modular graph is a concept related to the idea of module or modularity in the context of substructures of a graph. The term "modular graph" can sometimes be used in discussions of modular decomposition, which is a technique for breaking down a graph into simpler components based on the concept of modules.

A **multipartite graph** is a specific type of graph used in graph theory, where the vertex set can be divided into multiple distinct subsets such that no two vertices within the same subset are adjacent. In other words, the edges of the graph only connect vertices from different subsets.

Chris Godsil is an American mathematician known for his work in the fields of algebra, combinatorics, and quantum computing. He is particularly recognized for his contributions to graph theory and representation theory. Godsil has authored or co-authored numerous papers and has also been involved in teaching and mentoring students in these areas of mathematics.

Graph drawing is a field of computational geometry and computer science that focuses on the visualization of graphs, which are mathematical structures consisting of nodes (or vertices) connected by edges (or arcs). This area of study is concerned with developing algorithms and techniques to represent these graphs in a visually appealing and meaningful way, so that their structure and properties can be understood more easily.

Alfred Kempe (1935–2015) was an influential mathematician known primarily for his work in graph theory and combinatorics. He is perhaps best known for the Kempe Chains and the Kempe Conjecture, which are significant in the context of the Four Color Theorem, stating that any planar map can be colored with no more than four colors such that no two adjacent regions share the same color.

As of my last knowledge update in October 2021, Amanda Chetwynd is not a widely recognized public figure, and there may not be significant publicly available information about her. It's possible that she could be a private individual or a professional in a specific field that has not gained widespread recognition.

Arthur Hobbs is a mathematician known for his work in the field of mathematics, particularly in areas related to analysis and topology. However, there might not be extensive widely available information on his specific contributions or biography, as he may not be as widely recognized as some other figures in the field.

"Daniel Kráľ" could refer to a person, as it's a common name in some Slavic countries, particularly in Slovakia and the Czech Republic. However, without additional context, it is difficult to pinpoint exactly who or what you might be referring to. In general, "Daniel" is a common first name, and "Kráľ" translates to "king" in Slovak and Czech.

Carsten Thomassen is a Danish mathematician known for his contributions to graph theory and combinatorics. He has made significant advancements in various areas, particularly in the study of graph properties, graph coloring, and planar graphs. Thomassen is also recognized for his work on the relationships between graph theory and topology. He has published numerous papers and has been involved in various mathematical collaborations.

Frank Harary was an influential American mathematician known primarily for his contributions to the fields of graph theory and topology. He was born on May 8, 1921, and passed away on December 18, 2005. Harary is particularly well-known for introducing and popularizing various concepts in graph theory, including the study of networks and their applications.

Gabriel Andrew Dirac does not seem to be a widely recognized public figure, concept, or term as of my last knowledge update in October 2023. It's possible that he may be a private individual or a more recent figure who has gained prominence after that date.

Nick Wormald is a well-known figure in the fields of science and education, particularly recognized for his contributions to biology, ecology, and conservation. He has been involved in various research projects and educational initiatives aimed at promoting understanding of environmental issues and the importance of biodiversity.

Hassler Whitney was an influential American mathematician known for his contributions to topology and related areas of mathematics. He is particularly recognized for his work in the field of algebraic topology and for his contributions to the development of various mathematical theories and concepts. One of his notable contributions is the Whitney embedding theorem, which addresses how manifolds can be embedded into Euclidean spaces. This work has had significant implications in both pure mathematics and applied fields.